Notes on Polygon Meshes 1 Basic Definitions

CMSC 23700 Introduction to Computer Graphics Handout 2 Autumn 2015 November 12

Notes on polygon meshes

1 Basic deﬁnitions

Deﬁnition 1 A polygon mesh (or polymesh) is a triple (V,E,F ), where

V a set of vertices (points in space) E ⊂ (V × V ) a set of edges (line segments) F ⊂ E∗ a set of faces (convex polygons) with the following properties:

1. for any v ∈ V , there exists (v1, v2) ∈ E such that v = v1 or v = v2. 2. for and e ∈ E, there exists a face f ∈ F such that e is in f.

3. if two faces intersect in space, then the vertex or edge of intersection is in the mesh.

If all of the faces of a polygon mesh are triangles, then we call it a triangle mesh (trimesh). Polygons can be tessellated to form triangle meshes.

Deﬁnition 2 We classify edges in a mesh based on the number of faces they are part of:

• A boundary edge is part of exactly one face.

• An interior edge is part of two or more faces.

• A manifold edge is part of exactly two faces.

• A junction edge is part of three or more faces.

Junction edges are to be avoided; they can cause cracks when rendering the mesh.

Deﬁnition 3 A polymesh is connected if the undirected graph G = (VF ,EE), called the dual graph, is connected, where

• VF is a set of graph vertices corresponding to the faces of the mesh and

• EE is a set of graph edges connecting adjacent faces.

Deﬁnition 4 A polyhedron is a polymesh that is 1. connected and

2. each edge is manifold.

3 Deﬁnition 5 A polytope is a polyhedron that encloses a convex region R of R (i.e., any two points in R are connected by a line segment that is wholly contained in R).

Deﬁnition 6 A connected mesh is manifold if every edge in the mesh is either a boundary edge or a manifold edge.

For most computer graphic applications, we use manifold meshes.

Deﬁnition 7 A manifold mesh is closed if every edge is manifold and it is non-intersecting.

2 Orientation

The orientation of a face determines which side is the front and which side is the back. The orientation can either be Counter Clockwise (CCW, which is the OpenGL default) or Clockwise (CW).

Deﬁnition 8 Two faces, f1 and f2, that share a common edge e are consistently oriented if the head of e in f1 is the tail of e in f2 (and vice versa).

Deﬁnition 9 A manifold mesh is orientable if the vertex orderings of its faces can be chosen so that adjacent faces have consistent orderings (i.e., all faces are either CW or CCW).

3 Data structures

The data structures used to represent meshes vary by application. For rendering purposes, one might use a wireframe representation (just vertices and edges), or a trimesh representation (vertices and triangles). If we want to do more substantial computation with the mesh, we need to be able to efﬁciently answer geometric queries, such as

• ﬁnd the edges or vertices of a face

• ﬁnd the neighboring vertices of a vertex

• ﬁnd the faces of an edge

• ﬁnd the edges of a vertex (c.f., silhouette edges)

• ﬁnd the next edge in a path around a face

2 3.1 Winged-edge model

One popular representation is the winged-edge data structure. In this representation, the edge is the central part of the representation.

left succ right pred

left right face face

left right pred succ

In C, we might use the following pointer-based representation: struct edge { struct vertex *v0; /* endpoints of edge */ struct vertex *v1; struct face *left; /* face on left-hand-side of edge */ struct face *right; /* face on right-hand-side of edge */ struct edge *lPred; /* left-most predecessor edge */ struct edge *rPred; /* right-most predecessor edge */ struct edge *lSucc; /* left-most successor edge */ struct edge *rSucc; /* right-most successor edge */ }; struct vert { struct edge *e; ... }; struct face { struct edge *e; ... };

In a graphical application, we will store other information with vertices and faces (colors, normals, texture coordinates, ...), hence the “...” in the code. For models where the mesh is static, we can use a table-based representation that is more compact (assuming that we can use the char or short type as table indices).

3 3.2 Directed-edge model

The directed-edge (or half-edge) model splits the representation of each edge into two oriented parts. Consider the following fragment of a mesh:

b c

L el e er R

a d

The half-edge representation uses two half-edges (el and er) to represent the edge e. For el, we record the source vertex (u), the face to the left (L), the next half-edge in the CCW tour of L (e1), and the other half of the edge (er). Likewise, for er we record v, R, e2, and el. For a vertex we record an edge that it is a source of, and for a face we record an edge that the face lies to the left of. The C data structures for half-edges look like the following: struct Edge { Vertex *vert; // source vertex of edge Face *face; // face to left of edge Edge *next; // next edge in tour around face Edge *pair; // half-edge going the other way }; struct Vertex { Edge *edge; // An edge with this vertex as its source vec3f v; // The position of the vertex ... }; struct Face { Edge *edge; // An edge of the face ... };

In this representation, boundary edges will have a null pair pointer. When restricted to triangle meshes, the directed-edge model can be made very space efﬁcient (at a slight cost in time). Since each half-edge belongs to exactly one triangle, we can group them in triplets and use arithmetic to determine the triangle of an edge, the edges of a triangle, and the

4 next and previous edges of a tour:

tri(e) = e div 3 edges(t) = (3t, 3t + 1, 3t + 2) e + 2 if e mod 3 = 0 prev(e) = e − 1 otherwise e − 2 if e mod 3 = 2 next(e) = e + 1 otherwise

The edge representation can then become very compact: struct Edge { struct Vertex *vert; // source vertex of edge int pair; // half-edge going the other way };